and for the normalization constant \(\kappa ^{L}_{n,\alpha }\)  given in the Table REF , which fulfills (keep in mind that \(\Gamma (z) \sim e^{-z}\,z^z\,\left(\frac{2\pi }{z} \right)^{1/2}\) , see Eq. 5.11.3 of [1]})
\(\kappa ^{L}_{n,\alpha } \sim \kappa ^{L}_{n,\infty } \equiv \frac{\sqrt{2\pi }}{n!} \left(\frac{\alpha }{e}\right)^{\alpha }\,\alpha ^{n+1/2}, \qquad \alpha \rightarrow \infty ,\) 
